Constructivism, Realizability, Computability



The idea of realizability is a way of making the Brouwer-Heyting-Kolmogorov interpretation of constructivism and intuitionistic mathematics precise. It is related to the propositions as types paradigm. For instance, constructively a proof of an existential quantification xXϕ(x)\underset{x\in X}{\exists} \phi(x) consists of constructing a specific xXx \in X and a proof of ϕ(x)\phi(x), which “realizes” the truth of the statement, whence the name (see e.g. Streicher 07, section 1, Vermeeren 09, section 1 for introductions to the rough idea, or Bauer 05, page 12 and def. 4.7 for an actual definition).


type I computabilitytype II computability
typical domainnatural numbers \mathbb{N}Baire space of infinite sequences 𝔹= \mathbb{B} = \mathbb{N}^{\mathbb{N}}
computable functionspartial recursive functioncomputable function (analysis)
type of computable mathematicsrecursive mathematicscomputable analysis, Type Two Theory of Effectivity
type of realizabilitynumber realizabilityfunction realizability
partial combinatory algebraKleene's first partial combinatory algebraKleene's second partial combinatory algebra


Realizability originates with the interpretation of intuitionistic number theory, later developed as Heyting arithmetic, in

  • Stephen Kleene, On the interpretation of intuitionistic number theory Journal of Symbolic Logic, 10:109–124, 1945. link

A historical survey of realizability (including categorical realizability) is in

A quick survey is in

being a summary of

  • Martin Hyland, The effective topos, in The L.E.J. Brouwer Centenary

    Symposium (A. S. Toelstra and D. van Dalen, eds.), North-Holland Publishing Company, 1982, pp. 165–216

A modern textbook account is

  • Jaap van Oosten, Realizability: an introduction to its categorical side, Studies in Logic and the Foundations of Mathematics, vol. 152, Elsevier, 2008 (preface pdf)

Lecture notes include


  • Andrej Bauer, Realizability as connection between constructive and computable mathematics, in T. Grubba, P. Hertling, H. Tsuiki, and Klaus Weihrauch, (eds.) CCA 2005 - Second International Conference on Computability and Complexity in Analysis, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. (pdf)

based on

  • Andrej Bauer, The Realizability Approach to

    Computable Analysis and Topology_, PhD thesis CMU (2000) (pdf)

  • Peter Lietz, From Constructive Mathematics to Computable Analysis via the Realizability Interpretation (pdf.gz)

See also

Last revised on December 10, 2020 at 10:41:05. See the history of this page for a list of all contributions to it.